Multiplicity (statistical mechanics) explained

In statistical mechanics, multiplicity (also called statistical weight) refers to the number of microstates corresponding to a particular macrostate of a thermodynamic system.^[1] Commonly denoted

\Omega , it is related to the configuration entropy of an isolated system^[2] via Boltzmann's entropy formula$$S\; =\; k\_\backslash text\; \backslash log\; \backslash Omega,$$where S is the entropy and k _{B} is the Boltzmann constant.

Example: the two-state paramagnet

A simplified model of the two-state paramagnet provides an example of the process of calculating the multiplicity of particular macrostate.^[1] This model consists of a system of microscopic dipoles which may either be aligned or anti-aligned with an externally applied magnetic field . Let

N _{\uparrow} represent the number of dipoles that are aligned with the external field and N _{\downarrow} represent the number of anti-aligned dipoles. The energy of a single aligned dipole is U _{\uparrow} = - \mu B, while the energy of an anti-aligned dipole is U _{\downarrow} = \mu B; thus the overall energy of the system is$$U\; =\; (N\_\backslash downarrow-N\_\backslash uparrow)\backslash mu\; B.$$

The goal is to determine the multiplicity as a function of ; from there, the entropy and other thermodynamic properties of the system can be determined. However, it is useful as an intermediate step to calculate multiplicity as a function of

N _{\uparrow} and N _{\downarrow.} This approach shows that the number of available macrostates is . For example, in a very small system with dipoles, there are three macrostates, corresponding to N _{\uparrow=0,} 1 , 2 . Since the N _{\uparrow} = 0 and N _{\uparrow} = 2 macrostates require both dipoles to be either anti-aligned or aligned, respectively, the multiplicity of either of these states is 1. However, in the N _{\uparrow} = 1 , either dipole can be chosen for the aligned dipole, so the multiplicity is 2. In the general case, the multiplicity of a state, or the number of microstates, with N _{\uparrow} aligned dipoles follows from combinatorics, resulting in$$\backslash Omega\; =\; \backslash frac\; =\; \backslash frac,$$where the second step follows from the fact that N _{\uparrow+N} _{\downarrow} = N.

Since

N _{\uparrow} - N _{\downarrow} = - \tfrac{U}{\mu B}, the energy can be related to N _{\uparrow} and N _{\downarrow} as follows:

Thus the final expression for multiplicity as a function of internal energy is$$\backslash Omega\; =\; \backslash frac.$$

This can be used to calculate entropy in accordance with Boltzmann's entropy formula; from there one can calculate other useful properties such as temperature and heat capacity.

Notes and References

1. Book: Schroeder . Daniel V. . An Introduction to Thermal Physics . 1999 . Pearson . 9780201380279 . First.
2. Book: Atkins, Peter. Julio de Paula. 2002. Physical Chemistry. 7th. Oxford University Press.